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32) this leads us to an expression for the x-drift as 2b(x) = ∂x Σ(x,x) + Σ(x,x) −1 + z ν−1 + x2 x + ∂z Σ(x,z) + Σ(x,z) −1 x . 34) and similarly for the z-drift 2b(z) = ∂x Σ(x,z) + Σ(z,x) −1 + ν−1 z + 2 x x + ∂z Σ(z,z) + Σ(z,z) −1 x . 35) Thus, analogous to the one-dimensional case, we have an explicit mapping from Σ(·,·) to b(·) provided by the joint distribution P . 29), the methods we have developed would apply equally well to any abstract joint distribution with suitable decay properties, and may therefore also be of pure mathematical interest in their own right.

16) is N (N ) dEt = j=1 N (j) 1 (j) (j) (j) idϕt − dϕt 2 exp iϕt . 19) consists of a sum of independent randomly phased Wiener processes, with variance equal to BN dt, while the second term is independent of the scatterer label j. 20) where ξt is a complex Wiener process satisfying |dξt |2 = dt, dξt2 = 0. The process ξt is adapted to the filtration F (ϕ) = j F (j) , where F (j) is the filtration (j) appropriate to the component scatterer phase ϕt . 21) where the continuous valued random variable x, the average scattering ‘power’, ¯ .

Our aim is now to calculate the SDE satisfied by Bt , according to the rules of Ito calculus. From Ito’s formula we have   −1/2 n n 1 (n) (i) 2 (j)  W d Wt 2  dBt = 2 i=1 t j=1 − −3/2 n 3 1 2   (i) 2 Wt i=1 2 n (j) 2 d Wt  . 6) j=1 Now let us examine the term of the form d W 2 occurring above in isolation. 9 (n) Remarks. Concerning positivity. Directly from its definition, Bt is everywhere positive (or zero); this is reflected in its SDE by the divergence of the drift as zero is approached, which property (in spite of the persistence of the Wiener fluctuating term) ‘repels’ the process from the origin.

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Theoretical physics, - Electromagnetism and optics by Wilson W.

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